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Latitude by the stars

How do you find your way around the world? GPS? Well what if you
haven't got reception? A map? Good idea, but what if you're travelling
on the high seas with no land mark in sight? That's a situation that
many, many people have found themselves in over the millennia. These
brave seafarers used the Sun and the stars to navigate instead. And to do
this, they needed a fair bit of geometry, in particular
trigonometry.

Suppose you are on the open ocean and you want to work out your
latitude (see here for a definition of latitude). The Sun and most of the stars change their position
in the sky over time. But some stars always appear to be in the same
place. An example is Polaris, also called the North
star, which always appears to be sitting directly overhead the
North pole. It turns out that your latitude is the angle at which
Polaris appears to sit above the horizon.

To see why, let’s look at a two-dimensional picture. Consider the plane that contains the North pole, the point you are sitting at and the centre of the Earth. Strictly speaking, Polaris doesn’t sit vertically above , as shown in the picture, but it is so far away that the line of sight from to Polaris is very nearly vertical, so we can pretend that it does.

The angle that "Polaris sits above the horizon" is the angle indicated in the figure. It’s the angle our line of sight to Polaris, call it , forms with the line that is tangent to the Earth at the point (that’s our line of sight towards the horizon).

Extending and , we see the angle again on the other side of the crossing point :

The latitude of is defined to be the angle that the line from to makes with the plane containing the equator. In our two-dimensional picture, that equatorial plane is just a horizontal line which passes through It meets the vertical line in a point and the tangent line in a point

Because is the radius of the circle and a tangent, we know that and form a right angle at And since and form an angle we know that the angle between and is

Now consider the triangle with vertices , and As we have just seen, the angle at is Since is vertical and horizontal, the angle at is . And because angles in a triangle always add up to , the angle , which is our latitude, is

Et voilà — your latitude is given by the angle Polaris sits above the horizon. The Greek astronomer Hipparchus defined latitude in this way over 2000 years ago. He didn’t even know that the Earth was round, but our little picture here should explain why Hipparchus’ definition agrees with the modern one.

There is no equivalent of Polaris in the South, but to find your latitude if you are in the Southern hemisphere you can use a constellation called the Southern cross (illustrated on the flag of Australia) and two stars called the Southern Pointers.

Over the millennia navigators have used different devices to measure the angle at which a star appears above the horizon. These include beautiful astrolabes and sextants, which you often find in antique shops and museums.

That sorts out latitude, but how do you work out your longitude?
That's another story.